Correctors for Some Asymptotic Problems
Michel Chipota and Senoussi GuesmiaaReceived March 2009
Abstract—In the theory of anisotropic singular perturbation boundary value problems, the solution uεdoes not converge, in the H1-norm on the whole domain, towards some u0. In this
paper we construct correctors to have good approximations of uεin the H1-norm on the whole
domain. Since the anisotropic singular perturbation problems can be connected to the study of the asymptotic behaviour of problems defined in cylindrical domains becoming unbounded in some directions, we transpose our results for such problems.
DOI: 10.1134/S0081543810030211
1. INTRODUCTION
LetO = (−1, 1) × ω be a bounded open subset of Rp+1, p≥ 1, ω being a bounded open subset of Rp. We denote by x = (X
1, X2) the points ofO with
X1 = x1, X2 = (x1, . . . , xp).
With this notation we set
∇u =∂x1u, ∂x1u, . . . , ∂xpu T = ∂X1u ∇X2u , where ∇X2u = ∂x 1u, . . . , ∂xpu T .
For f ∈ L2(O) and ε > 0, there exists a unique solution uε (in a weak sense) of
uε∈ H01(O), −ε2∂2
X1uε− ΔX2uε= f in O.
(1.1)
We denote by ΔX2 the Laplace operator defined by ΔX2 = ∂
2
x1 + . . . + ∂
2
xp. For a.e. X1∈ (−1, 1) one can define a solution u0 to
u0(X1,·) ∈ H01(ω), −ΔX2u0(X1,·) = f(X1,·) in ω. (1.2) It is shown in [3, 4] that uε → u0 in L2(O) as ε → 0. (1.3)
aInstitute of Mathematics, University of Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland.
E-mail addresses: m.m.chipot@math.uzh.ch (M. Chipot), senoussi.guesmia@math.uzh.ch (S. Guesmia). 263
Even if ∇X2uε→ ∇X2u0 in (L
2(O))p (see [3, 4]), one cannot expect in general that
uε→ u0 in H1(O). (1.4)
Indeed, if, for instance, f is independent of X1, then so is u0 and clearly, for f = 0, u0 ∈ H/ 01(O)
when uε does belong to H01(O), which makes (1.4) impossible. The goal of this paper is to “correct” uε−u0 by a simple function wεwhich gives the behaviour of uε−u0near the end sections{−1, 1}×ω
and is such that
uε− u0− wε→ 0 in H01(O). (1.5)
Due to the uniqueness of a solution of (1.1), one has (see Lemma 3.4)
uε(−X1, X2) = uε(X1, X2),
and this clearly implies that
∂uε
∂X1
(0, X2) = 0. (1.6)
Thus, to study and correct the behaviour of uε− u0, one can consider uε as the solution to
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −ε2∂2 X1uε− ΔX2uε = f in Ω = (0, 1)× ω, uε= 0 on ∂Ω\ {0} × ω, ∂uε ∂X1 = 0 on {0} × ω. (1.7)
This is what we will do in the next section. Note that this is inspired by [5], where a similar analysis was carried out for the Stokes problem. In Section 3 we will transpose our results—via a scaling argument—to the Dirichlet problem set in cylinders becoming infinite in various directions.
For more details about the anisotropic singular perturbation problems, as well as for details on the problems considered in Section 3, we refer the reader to [1–6, 8, 9]. The classic singular perturbation problems are dealt with in [10].
2. THE CASE OF ANISOTROPIC PROBLEMS IN ONE DIRECTION Let Ω be defined as
Ω = (0, 1)× ω, where ω is a bounded domain of Rp, and V be the space
V = v∈ H1(Ω)| v = 0 on ∂Ω \ {0} × ω . (2.1)
There exists a unique solution uε to
⎧ ⎪ ⎨ ⎪ ⎩ uε ∈ V, Ω ε2∂X1uε∂X1v +∇X2uε· ∇X2v dx = Ω f v dx ∀ v ∈ V. (2.2)
Clearly (2.2) is the weak formulation of (1.7). We assume that f ∈ L2(Ω), and the existence of a unique solution to (2.2) follows from the Lax–Milgram theorem. The weak formulation of (1.2) reads for a.e. X1∈ (0, 1) as
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u0(X1,·) ∈ H01(ω), ω ∇X2u0(X1,·) · ∇X2v dX2 = ω f (X1,·)v dX2 ∀ v ∈ H01(ω). (2.3)
In the case where
f ∈ L2(Ω), ∂X1f ∈ L
2(Ω), (2.4)
one can show (see [4]) that
u0 ∈ H1(Ω).
Now (see, for instance, [2]) if v∈ V , then for a.e. X1∈ (0, 1) one has
v(X1,·) ∈ H01(ω). (2.5)
Using this test function in (2.3), one derives, after an integration in X1, that
Ω ∇X2u0· ∇X2v dx = Ω f v dx ∀ v ∈ V. (2.6)
In order to construct a corrector for uε, we denote by S the half-cylinder
S = (, +∞) × ω,
where ∈ R. Then if ρ: [0, +∞) → [0, 1] is the function defined by
ρ(x) =
1− x on [0, 1], 0 on (1, +∞), we introduce u as the solution to
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u∈ H01(S0), S0 ∇u · ∇v dx = S0 ∇(ρ(X1)u0)· ∇v dx ∀ v ∈ H01(S0). (2.7)
Since u0∈ H1(Ω), the existence and uniqueness of u follows from the Lax–Milgram theorem. Then
we set
w(X1, X2) = u(X1, X2)− ρ(X1)u0(X1, X2) = u− ρu0 (2.8)
and denote by wε the function defined as
wε(X1, X2) = w 1− X1 ε , X2 . (2.9) Note that w∈ H1(S
0) and satisfies in a weak sense
Δw = 0 in S
0,
w =−u0 on {0} × ω, w = 0 on (0, +∞) × ∂ω.
(2.10) 2.1. Some preliminary results. We denote by Ω− the domain defined by
Ω−= (−1, 0) × ω. For v∈ V we define by v the function given by
v(X1, X2) =
v(X
1, X2), X1 ≥ 0, v(−X1, X2), X1 < 0.
Then we have
Lemma 2.1. For every v∈ V the following equality holds: Ω ε2∂X1wε∂X1v +∇X2wε· ∇X2v dx =− Ω− ε2∂X1wε∂X1v + ∇X2wε· ∇X2v dx.
Proof. For > 0 we set Ω = (0, )×ω. Then first note that for v ∈ V we have v(1−εX1, X2)∈ H1
0(Ω2/ε). Thus we derive, from (2.10),
Ω2/ε ∇w · ∇v(1 − εX1, X2) dx = 0, whence Ω1/ε ∇w · ∇v(1 − εX1, X2) dx =− Ω2/ε\Ω1/ε ∇w · ∇v(1 − εX1, X2) dx. (2.12)
Making the change of variable X1 = 1− εX1 in the integrals above, we obtain respectively
Ω1/ε ∇w · ∇v(1 − εX1, X2) dx = 1 ε Ω ε2∂X1wε∂X1v +∇X2wε· ∇X2v dx and Ω2/ε\Ω1/ε ∇w · ∇v(1 − εX1, X2) dx = Ω2/ε\Ω1/ε −ε∂X1w ∂X1v(X1, X2) +∇X2w· ∇X2v(X1, X2) dx = 1 ε Ω− ε2∂X1wε∂X1v + ∇X2wε· ∇X2v dx.
The lemma follows from (2.12).
We will also need the following lemma.
Lemma 2.2. There exist positive constants C > 0 and α > 0 independent of ε such that S1/ε |∇w|2dx≤ Ce−αε S0 |∇w|2dx. (2.13)
Proof. Without loss of generality, we assume that ε < 1. Let γε: R → R be a continuous
function such that γε = 0 in
−∞,1 ε − 1 , γε = 1 in 1 ε, +∞ and γε is linear in 1 ε − 1, 1 ε . Since γε(X1)w∈ H01(S0), we have, by (2.10), S0 ∇w · ∇(γε(X1)w) dx = 0. (2.14) Thus S0 |∇w|2γ ε(X1) dx =− S1/ε−1\S1/ε ∂X1w ∂X1γε(X1)w dx≤ S1/ε−1\S1/ε |∂X1w| |w| dx ≤ 1 2 S1/ε−1\S1/ε |∂X1w| 2dx +1 2 S1/ε−1\S1/ε |w|2dx.
Applying the Poincar´e inequality in X2 to the last integral, we get for some constant Cω S1/ε−1\S1/ε |w|2dx = 1/ε 1/ε−1 ω |w|2dx≤ C ω 1/ε 1/ε−1 ω |∇X2w| 2dx. This leads to S1/ε |∇w|2dx≤ max(1, Cω) 2 S1/ε−1\S1/ε |∇w|2dx = max(1, Cω) 2 S1/ε−1 |∇w|2dx−max(1, Cω) 2 S1/ε |∇w|2dx, and thus S1/ε |∇w|2dx≤ r S1/ε−1 |∇w|2dx, where r = max(1,Cω) 2+max(1,Cω). Iterating 1 ε
times this formula (1εis the integer part of 1ε), we obtain S1/ε |∇w|2dx≤ r[1ε] S1/ε−[1/ε] |∇w|2dx. Since 1ε − 1 <1ε≤ 1ε, we deduce S1/ε |∇w|2dx≤ 1 re ln r1 ε S0 |∇w|2dx.
This completes the proof by setting C = 1r and α =− ln r.
The theorem below will play an important role in the following.
Theorem 2.3. Let uε and u0 be the solutions to (1.7) and (1.2), respectively. Then under the assumption (2.4) there exist two constants C and α > 0 independent of ε, such that
3 4 Ω ε2(∂X1(uε− u0− wε)) 2+|∇ X2(uε− u0− wε)| 2dx ≤ Ce−α ε S0 |∇w|2dx− ε2 Ω ∂X1u0∂X1(uε− u0− wε) dx. (2.15)
Proof. Subtracting (2.6) from (2.2), we obtain Ω ε2∂X1(uε− u0)∂X1v +∇X2(uε− u0)· ∇X2v dx =−ε2 Ω ∂X1u0∂X1v dx ∀ v ∈ V. (2.16) Since uε− u0− wε∈ V , we get Ω ε2∂X1(uε− u0)∂X1(uε− u0− wε) +∇X2(uε− u0)· ∇X2(uε− u0− wε) dx =−ε2 Ω ∂X1u0∂X1(uε− u0− wε) dx.
According to Lemma 2.1, the identity above can be written as Ω ε2(∂X1(uε− u0− wε)) 2+|∇ X2(uε− u0− wε)| 2dx =− Ω ε2∂X1wε∂X1(uε− u0− wε) +∇X2wε· ∇X2(uε− u0− wε) dx − ε2 Ω ∂X1u0∂X1(uε− u0− wε) dx = Ω− ε2∂X1wε∂X1(uε− u0− wε) +∇X2wε· ∇X2(uε− u0− wε) dx − ε2 Ω ∂X1u0∂X1(uε− u0− wε) dx. (2.17)
We separately estimate the integral over Ω− using the Cauchy–Schwarz and Young’s inequalities. We derive Ω− ε2∂X1wε∂X1(uε− u0− wε) +∇X2wε· ∇X2(uε− u0− wε) dx ≤ ⎛ ⎝ Ω− ε2(∂X1wε) 2+|∇ X2wε| 2dx ⎞ ⎠ 1/2 × ⎛ ⎝ Ω− ε2(∂X1(uε− u0− wε)) 2+|∇ X2(uε− u0− wε)| 2dx ⎞ ⎠ 1/2 ≤ Ω− ε2(∂X1wε) 2+|∇ X2wε| 2dx +1 4 Ω ε2|∂X1(uε− u0− wε)| 2+|∇ X2(uε− u0− wε)| 2dx.
Going back to (2.17), we find that 3 4 Ω ε2(∂X1(uε− u0− wε)) 2+|∇ X2(uε− u0− wε)| 2dx ≤ Ω− ε2(∂X1wε) 2+|∇ X2wε| 2dx− ε2 Ω ∂X1u0∂X1(uε− u0− wε) dx.
Making the change of variable X1→ 1−Xε 1 in the first integral of the second line, we get
3 4 Ω ε2(∂X1(uε− u0− wε)) 2+|∇ X2(uε− u0− wε)| 2dx ≤ ε Ω2/ε\Ω1/ε |∇w|2dx− ε2 Ω ∂X1u0∂X1(uε− u0− wε) dx ≤ ε S1/ε |∇w|2dx− ε2 Ω ∂X1u0∂X1(uε− u0− wε) dx. (2.18)
Combining (2.13) and (2.18) leads to the basic inequality (2.15). This completes the proof of the theorem.
2.2. Convergence results. As a first application of Theorem 2.3 we have
Theorem 2.4. The solution u0 is a strong limit of the sequence uε− wε in H1(Ω) and the
following error estimate is valid :
|uε− u0− wε|L2(Ω),|∇X2(uε− u0− wε)|L2(Ω)= o(ε),
|∂X1(uε− u0− wε)|L2(Ω)= o(1).
Proof. Applying the Cauchy–Schwarz inequality to the last term of (2.15), we derive 3 4 Ω ε2(∂X1(uε− u0− wε)) 2+|∇ X2(uε− u0− wε)| 2dx ≤ Ce−αε S0 |∇w|2dx + ε2|∂ X1u0|L2(Ω)|∂X1(uε− u0− wε)|L2(Ω).
Then by Young’s inequality we get for some constant C
ε2|∂X1(uε− u0− wε)| 2 L2(Ω)+|∇X2(uε− u0− wε)| 2 L2(Ω)≤ Ce− α ε S0 |∇w|2dx + Cε2|∂ X1u0| 2 L2(Ω). This estimate shows in particular that
|∇X2(uε− u0− wε)|L2(Ω)= O(ε) (2.19) since e−αε = o(ε2). At the same time we have proved the boundedness of |∂X
1(uε− u0− wε)|L2(Ω). This allows us to extract a weakly convergent subsequence of ∂X1(uε − u0 − wε) in L
2(Ω) and
according to (2.19) it follows that the whole sequence converges weakly to 0, i.e.
∂X1(uε− u0− wε) 0 in L
2(Ω).
Going back to (2.15), using the fact that e−αε = o(ε2) and the weak convergences above, we obtain
|∇X2(uε− u0− wε)|L2(Ω)= o(ε), |∂X1(uε− u0− wε)|L2(Ω) = o(1).
Finally, using the Poincar´e inequality in the direction X2, with the help of the estimates above we
complete the proof of the theorem.
We can improve the rate of convergence above if we assume more smoothness of f as in the following theorem.
Theorem 2.5. Under the assumptions of Theorem 2.3 and if
∂X21u0 ∈ L2(Ω) and ∂X1u0 = 0 on {0} × ω, (2.20)
we have
|uε− u0− wε|L2(Ω),|∇X2(uε− u0− wε)|L2(Ω)= O(ε2),
|∂X1(uε− u0− wε)|L2(Ω)= O(ε)
Remark 2.6. (i) The second hypothesis in (2.20) means that for a.e. X2∈ ω we have ∂X1u0(0, X2) = 0.
(ii) For instance, if f is smooth enough, we can show that the hypotheses
∂X21f ∈ L2(Ω) and ∂X1f = 0 on {0} × ω imply (2.20) using the representation formula
u0(x) =
ω
f (X1, y)G(X2, y) dy,
where G is the Green function (see [7]).
Proof of Theorem 2.5. Integrating by parts the last integral of (2.15), we get 3 4 Ω ε2(∂X1(uε− u0− wε)) 2+|∇ X2(uε− u0− wε)| 2dx ≤ Ce−αε S0 |∇w|2dx + ε2 Ω ∂X21u0(uε− u0− wε) dx + ε2 ω ∂X1u0(0, X2)(uε− u0− wε)(0, X2) dX2 = Ce−αε S0 |∇w|2dx + ε2 Ω ∂X21u0(uε− u0− wε) dx
(uε− u0− wε∈ V and ∂X1u0 = 0 on{0} × ω). By the Cauchy–Schwarz and Young’s inequalities it follows that 3 4 Ω ε2(∂X1(uε− u0− wε)) 2+|∇ X2(uε− u0− wε)| 2dx ≤ Ce−α ε S0 |∇w|2dx + ε2|∂2 X1u0|L2(Ω)|uε− u0− wε|L2(Ω) ≤ Ce−α ε S0 |∇w|2dx + με4|∂2 X1u0| 2 L2(Ω)+ 1 4μ|uε− u0− wε| 2 L2(Ω) ≤ Ce−α ε S0 |∇w|2dx + με4|∂2 X1u0| 2 L2(Ω)+ Cω 4μ|∇X2(uε− u0− wε)| 2 L2(Ω),
where Cω is the Poincar´e inequality constant. Choosing μ = Cω and since e− α ε = o(ε4), we are ending up with ε2 Ω (∂X1(uε− u0− wε)) 2+|∇ X2(uε− u0− wε)| 2dx≤ Cε4.
Applying the Poincar´e inequality to uε− u0− wε∈ V , we complete the proof of the theorem.
Thanks to Theorem 2.3, if we assume that f is independent of X1, we get an exponential rate
Theorem 2.7. Under the assumptions above and if in addition f is independent of X1, we have an exponential convergence of uε− wε to u0 in the whole domain Ω, i.e. there exist positive constants C and α independent of ε such that
Ω |∇(uε− u0− wε)|2dx≤ Ce− α ε S0 |∇w|2dx.
Proof. This is an immediate consequence of (2.15).
3. PROBLEMS IN DOMAINS BECOMING UNBOUNDED Let Ωm be a bounded open subset of Rm+p defined by
Ωm =
(0, )m× ω if > 0,
(, 0)m× ω if < 0,
where m, p > 0 are integers and ω is a bounded open subset of Rp. For simplicity we drop the index 1 in Ω1 and Ω1; i.e., to be consistent with our notation of Section 1, we set
Ω1 := Ω, Ω1 := Ω.
The points inRm+p will be denoted by x = (X1, X2) = (x1, . . . , xm, x1, . . . , xp) with
X1 = (x1, . . . , xm), X2 = (x1, . . . , xp).
With this notation we set
∇u =∂x1u, . . . , ∂xmu, ∂x1u, . . . , ∂xpu T = ∇X1u ∇X2u , where ∇X1u = ∂x1u, . . . , ∂xmu T , ∇X2u = ∂x 1u, . . . , ∂xpu T .
We divide the boundary Γm of Ωm into two partsDm and Nm such that
Nm = i=1,...,m {xi = 0} ∩ Ωm , Dm = Γm \ Nm. We also set N1 :=N, N1 :=N , D1 :=D, D1 :=D.
In this section we deal with the asymptotic behaviour, when → +∞, of the solution um to the
Laplace boundary value problem ⎧ ⎪ ⎨ ⎪ ⎩ −Δum = f in Ωm , um = 0 on Dm, ∂ηum = 0 on Nm, (3.1)
where f is independent of X1, i.e.
f (x) = f (X2)∈ L2(ω).
Here ∂η denotes the derivative along the outward normal to the boundary Γm . The existence of a
weak solution um of (3.1) is ensured by the Lax–Milgram theorem in the space
Thanks to Lemma 3.1 below and [6, Theorem 1.1] it follows that um converges towards the so-lution u0 to (1.2), as → +∞, in H1(Ωm0) where 0 < is a constant. More precisely, we have
in fact
Ωm /2
|∇(um
− u0)|2dx≤ Ce−α, (3.2)
where C and α are positive constants independent of . In this section we are interested in the asymptotic behaviour of um in the neighbourhood of Dm. We start with the case m = 1 in the following subsection and next we consider the general case.
3.1. Domains becoming unbounded in one direction.
3.1.1. Mixed boundary value problems. We consider here the special case m = 1. By making the change of variable
X1→
1
εX1, (3.3)
where ε = 1, we deduce that u1 is a solution of (3.1) if and only if the function
Ω→ R, x→ u1
1
εX1, X2
is a solution of (1.7). Then we set
w(X1, X2) := wε 1 X1, X2 = w(− X1, X2),
where w is a solution of (2.10). The following theorem is a direct consequence of Theorem 2.7 and (3.3).
Theorem 3.1. We have the convergence u1−w → u0 on the whole domain Ω, i.e. in H01(Ω),
and the following estimate is true:
Ω |∇(u1 − u0− w)|2dx≤ Ce−α S0 |∇w|2dx, (3.4)
where C and α are positive constants independent of .
Remark 3.2. Estimate (3.2) is a corollary of Theorem 3.1. Indeed, we have Ω/2 |∇(u1 − u0)|2dx≤ 2 Ω/2 |∇(u1 − u0− w)|2dx + 2 Ω/2 |∇w|2dx ≤ Ce−α S0 |∇w|2dx + 2 S/2 |∇w|2dx,
by a change of variable. The last integral converges towards 0 at an exponential rate by Lemma 2.2, which shows (3.2).
Remark 3.3. For any a > 0,
Ω−a
|∇(u1
To show this, one notices that Ω−a |∇(u1 − u0)|2dx = Ω−a |∇(u1 − u0− w) +∇w|2dx ≥ 1 2 Ω−a |∇w|2dx− Ω−a |∇(u1 − u0− w)|2dx = 1 2 Sa |∇w|2dx + o(1).
Since w is a harmonic function, one has for every a
Sa
|∇w|2dx > 0.
Then the convergence of u1 towards u0 may not occur in H1(Ω).
3.1.2. Dirichlet boundary value problems. Let us consider in O = (−, ) × ω the Dirichlet
boundary value problem
−ΔU= f in O,
U = 0 on ∂O.
It is clear that U is a unique function of H01(O) satisfying
O ∇U· ∇v dx = O f v dx ∀v ∈ H01(O). (3.5)
The following lemma summarizes some useful properties of the solution U.
Lemma 3.4. Under the previous assumptions, we have
• U(−X1, X2) = U(X1, X2) for a.e. x∈ O;
• the restriction of U to Ω is a unique solution to
⎧ ⎪ ⎨ ⎪ ⎩ U ∈ V, Ω ∇U· ∇v dx = Ω f v dx ∀ v ∈ V.
Proof. For v ∈ H01(O) denote by v the function defined by v(X1, X2) = v(−X1, X2). It is
clear thatv ∈ H1
0(O), and if we make the change of variable X1 =−X1 in (3.5), we derive
O ∇ U· ∇v dx = O ∇ U· ∇vdx = O −∂X1U(−∂X1v) + ∇X2U· ∇X2v (X1, X2) dx = O ∇U· ∇vdx = O fvdx = O f v dx,
since f is independent of X1. This means that U is also a weak solution to (3.5) and by uniqueness
of the solution we deduce the first point of the lemma. For v∈ V we can easily check that v defined by (2.11) belongs to H1 0(O). Moreover, we have O ∇U· ∇vdx = Ω ∇U· ∇vdx + Ω− ∇U· ∇vdx.
Thanks to the first point, the last integral can be written as (X1 =−X1)
Ω− ∇U· ∇vdx = Ω− −∂X1U(−X1, X2)∂X1v(X1, X2) +∇X2U· ∇X2v dx = Ω ∇U· ∇v dx.
Thus we have for every v ∈ V
O ∇U· ∇vdx = 2 Ω ∇U· ∇v dx. (3.6) Also by (3.5) we have O ∇U· ∇vdx = O fvdx = 2 Ω f v dx. (3.7)
Combining (3.6) and (3.7) shows the second point.
As a consequence of the second point of Lemma 3.4, it follows that
U = u1 on Ω.
Then, thanks to Theorem 3.1 and the first point of Lemma 3.4 we can state
Theorem 3.5. There exist positive constants C and α independent of such that O |∇(U− u0− w)|2dx≤ Ce−α S0 |∇w|2dx.
3.2. More general domains. For m = 1, we defined in the previous subsection a corrector
w1 := w satisfying (3.4). In order to construct a corrector for m = 2, we introduce a function
w2 ∈ H1
0((0, +∞) × Ω1) as a solution to
Δw2 = 0 in (0, +∞) × Ω1,
w2 =−u0− w1 on {0} × Ω1, w2= 0 on (0, +∞) × ∂Ω1.
The existence of w2 is ensured by the Lax–Milgram theorem. The corrector candidate in this case is w1 + w2 where w2 is given by
w2(x1, x2, X2) = w2(− x1, x2, X2).
Instead of showing this only for the case m = 2, we construct by induction for i = 2, . . . , m functions
wi: S0i → R defined as follows. For a solution u to ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u∈ H01(S0i), Si 0 ∇u · ∇v dx = Si 0 ∇ ρ(x1) u0+ i−1 j=1 wj · ∇v dx ∀ v ∈ H1 0(S0i), (3.8)
where Sai = (a, +∞) × Ωi−1 (a∈ R), we set wi(x1, . . . , xi, X2) = u(x1, . . . , xi, X2)− ρ(x1) u0(X2) + i−1 j=1 wj(xi−j+1, . . . , xi, X2) (3.9)
and denote by wi the function defined as
wi(x) = wi(− x1, x2, . . . , xi, X2).
Then we have
Theorem 3.6. Under the assumptions above, the difference um −
m j=1w
j
converges
to-wards u0 on the whole domain Ωm , i.e. in H01(Ωm ), and there exist positive constants C and α
independent of such that
Ωm ∇ um − u0− m j=1 wj 2 dx≤ Ce−α. (3.10)
Proof. In order to check that mj=1wj(xm−j+1, . . . , xm, X2) is a corrector corresponding to
problem (3.1) and satisfying (3.10), we will argue by induction. According to Theorem 2.4 the statement holds when m = 1; then we assume that mj=1−1wj(xm−j+1, . . . , xm, X2) is a corrector
satisfying Ωm−1 ∇ um −1− u0− m−1 j=1 wj 2 dx≤ Ce−α, (3.11)
where C and α are some positive constants independent of . In the following we show the same estimate for m. Let us introduce a function wm defined as below. For a solution u to
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u∈ H01(S0m), Sm 0 ∇u · ∇v dx = Sm 0 ∇ρ(x1)um−1 · ∇v dx ∀ v ∈ H1 0(S0m), (3.12) we set w(x) = u(x)− ρ(x1)um −1(x2, . . . , xm, X2) (3.13)
(w depends on ) and denote by wm the function defined as
wm(x) = w(− x1, x2, . . . , xm, X2). (3.14)
Then we have
Lemma 3.7. For any > 0, there exist constants C > 0 and α > 0 such that
Ωm |∇(um − um −1− w m )|2dx≤ Ce−α . (3.15)
Proof. Without loss of generality, we assume that > 1. Arguing as in the previous section and replacing ω by Ωm −1, we can show an estimate similar to (3.4), i.e.
Ωm |∇(um − um −1− w m )|2dx≤ Ce−α Sm 0 |∇w|2dx. (3.16)
(We use the fact that Ωm −1 is bounded in the direction X2 to get a Poincar´e constant independent
of .) We have now to estimate the last integral in (3.16). By using, in (3.12), v = u we obtain easily |∇u|2 L2(Sm 0 )= Sm −1\Sm ∇ρ(x1)um −1 · ∇u dx ≤ C|∇um−1 |L2(Ωm−1 )|∇u|L 2(Sm 0 ), whence |∇u|L2(Sm 0 )≤ C|∇u m−1 |L2(Ωm−1 ). Then by (3.13) we derive |∇w|L2(Sm 0 )≤ |∇u|L2(Sm0 )+|∇(ρ(x1)u m−1 )|L2(Sm 0 )≤ C|∇u m−1 |L2(Ωm−1 ),
where C is independent of . Next, taking in the weak formulation of (3.1), written for m− 1,
v = um −1 yields |∇um−1 |L2(Ωm−1 )≤ C|f| 2 L2(Ωm−1 ) = Cm−1|f|2L2(ω), since f is independent of X1. Thus, it follows that
|∇w|L2(Sm 0 ) ≤ C
m−1|f|2
L2(ω). (3.17)
Going back to (3.16), we have Ωm |∇(um − um −1− w m )|2dx≤ Cm−1|f|2L2(ω)e−α.
Since → +∞, there exist constants 0 < α < α and C > 0 such that
Ωm |∇(um − um −1− w m )|2dx≤ Ce−α .
This completes the proof of the lemma.
We now return to the proof of the theorem. The integral in (3.10) can be estimated as Ωm ∇ um − u0− m j=1 wj 2 dx≤ 3 Ωm |∇(um − u m−1 − w m )|2dx + 3 Ωm ∇ um −1− m−1 j=1 wj− u0 2 dx + 3 Ωm |∇(wm − wm )|2dx.
The exponential convergences to 0 of the first and the second integral of the right-hand side are given by (3.15) and the induction hypothesis (3.11), respectively. Then it remains to show the same rate of convergence for the last integral to complete the proof. First, we estimate the difference between w and wm, defined in (3.13) and (3.9), respectively, as
|∇(w − wm)| L2(Sm 0 )≤ |∇(u − u)|L2(S0m)+ ∇ ρ(x1) um −1− u0− m−1 j=1 wj L2(Sm 0 ) . (3.18)
We estimate the last term in the inequality above using the Poincar´e inequality and the induction hypothesis (3.11); then we have
∇ ρ(x1) um −1− u0− m−1 j=1 wj L2(Sm 0 ) ≤ C ∇ um −1− u0− m−1 j=1 wj L2(Ωm−1 ) ≤ Ce−α. (3.19)
For the first term of the right-hand side of (3.18), we compare (3.12) and (3.8) for i = m− 1 and, taking v = u− u ∈ H01(S0m) as a test function, obtain
|∇(u − u)|2 L2(Sm 0 )≤ ∇ ρ(x1) um −1− u0− m−1 j=1 wj L2(Sm 0 ) |∇(u − u)|L2(Sm 0 ). Applying (3.19) here and in (3.18), we get
|∇(w − wm)| L2(Sm
0 )≤ Ce
−α. (3.20)
Finally, the change of variable x1 → − x1 and (3.20) lead to |∇(wm − wm )|L2(Ωm )=|∇(w − w m)| L2((0,2)×Ωm−1 ) ≤ |∇(w − wm)| L2(Sm 0 ) ≤ Ce−α.
This completes the proof.
Remark 3.8. As in Theorem 3.5, using symmetries, we can construct correctors for the Laplace equation defined in (−, )m× ω coupled with the homogeneous Dirichlet boundary conditions.
ACKNOWLEDGMENTS
The authors have been supported by the Swiss National Science Foundation under the contracts #20-113287/1 and #20-117614/1. They are very grateful to this institution.
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